Numerical Linear Algebra

Condition numbers measure how sensitive a mathematical function or system is to small changes in input. Learners will understand how to evaluate error propagation and assess the stability of numerical algorithms.

Eigenvalue problems involve finding vectors whose direction remains unchanged under a given linear transformation. Learners will understand how to compute eigenvalues and eigenvectors to analyze dynamical systems and structural stability.

Iterative methods generate sequences of approximations to solve complex mathematical problems, particularly large linear systems. Learners will understand convergence criteria, error bounds, and how to implement these algorithms for computational efficiency.

Krylov subspace methods are iterative algorithms used to find eigenvalues or solve large, sparse systems of linear equations. Learners will understand methods like GMRES and Conjugate Gradient to solve high-dimensional computational problems.

LU factorization decomposes a matrix into a product of a lower triangular matrix and an upper triangular matrix. Learners will understand how to apply this decomposition to solve linear systems efficiently.

Matrix computations involve the algorithmic manipulation of matrices to solve scientific and engineering problems. Learners will understand matrix multiplication, inversion, and decomposition techniques optimized for speed and numerical accuracy on computers.

Numerical methods are algorithms used to obtain approximate numerical solutions to mathematical problems that lack analytical solutions. Learners will understand how to solve differential equations, find roots, and perform numerical integration using computers.

Numerical stability describes how round-off errors and approximations propagate through an algorithm. Learners will understand how to analyze algorithm behavior to ensure computational results remain accurate and do not diverge.

QR factorization decomposes a matrix into an orthogonal matrix and an upper triangular matrix. Learners will understand how to compute this decomposition and apply it to solve least squares and eigenvalue problems.

Singular value decomposition factorizes a real or complex matrix into three constituent matrices. Learners will understand how to apply this technique for dimensionality reduction, data compression, and solving ill-posed linear systems.

Sparse matrices are matrices populated primarily with zeros, requiring specialized storage and computational techniques. Learners will understand how to store, manipulate, and solve linear systems involving these matrices efficiently.